The aim of this note is to deduce error estimates for a fully-discrete finite element method approximation of a kind of degenerate mixed parabolic equations. The obtained results consider regularity assumptions about the main variable according to the degenerate character of the problem, given by the term involving the time-derivative, which is represented with a non-invertible linear operator R. We show two different approaches to obtain the error estimates. The first one needs to introduce an extension operator of R and the second one requires to add a new ellipticity property for this operator. These error estimates can be applied to analyze the fully-discrete finite element method approximation of an eddy current model.